Donald Trump makes remarks as he hosts a naturalization ceremony in the Oval Office on Saturday.
Photograph: REX/Shutterstock

Donald Trump raised the possibility of one day granting amnesty to migrants living in the US illegally, after Democrats rejected his latest plan to fund a wall along the southern border and reopen the US government.

In a remark that angered Republicans while not being taken seriously by Democrats, Trump suggested legal status could be given to millions of undocumented people as part of a future grand bargain on American immigration law.

The president floated the idea in a tweet on Sunday that stressed he was, for now, only offering to extend legal protections for some refugees and people who were brought to the US illegally as children.

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“Amnesty will be used only on a much bigger deal, whether on immigration or something else,” Trump said. He also said there would “no big push” to deport those already living in the US without permission.

The US is more than four weeks into its longest government shutdown, which was triggered by Trump’s refusal to sign a bipartisan congressional spending plan that did not give him the billions of dollars he wants for a wall along the border with Mexico.

Trump’s emphatic promise to “build the wall” won him the support of many conservative voters during the 2016 presidential campaign. He claimed he would force Mexico to pay for the wall directly, but has recently effectively conceded this will not happen.

And by the way, clean up the streets in San Francisco, they are disgusting!

Donald Trump, to Nancy Pelosi

On Saturday, Trump said from the White House that he would agree to limited concessions for some undocumented immigrants if Democrats agreed to give him more than $5bn in public funds for the wall.

Trump’s offer would comprise a three-year extension in legal protection for roughly 700,000 “dreamers”, were brought to the US illegally as children, and approximately 300,000 refugees facing an end to temporary legal status.

The proposal was dismissed by senior Democrats even before Trump began speaking. Democrats are demanding that Trump reopen the government by signing the existing congressional spending plan before any further negotiations on immigration.

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Nancy Pelosi, the Democratic speaker of the House of Representatives, described Trump’s plan as a “compilation of several previously rejected initiatives” that would not provide lasting security for dreamers and other groups.

“What is original in the president’s proposal is not good,” she said. “What is good in the proposal is not original.”

Republicans in the Senate will nonetheless take up Trump’s proposals next week. The Oklahoma senator James Lankford told ABC’s This Week the offer was “a reasonable compromise” and said: “The vote this week is not to pass the bill. It’s to open up and say, ‘Can we debate this? Can we amend it? Can we make changes?’”

On Sunday, Schumer said he did not think the president’s measures would pass the Senate. He also said he would push legislation to shield government workers from eviction or home foreclosure, repossession of cars and penalties for late payment of bills and student loans.

Trump, meanwhile, insulted Pelosi and derided the cleanliness of the California city that she represents in Congress. He wrote in a tweet: “And by the way, clean up the streets in San Francisco, they are disgusting!”

The president’s proposal also met hostile reactions among many on the Republican right, highlighting a political dilemma that Trump, who claims to be a master dealmaker, has created for himself.

Some conservatives dismissed the offer as being akin to an amnesty itself. Ann Coulter, the far-right author, said in a tweet: “Trump proposes amnesty. We voted for Trump and got Jeb!”

The former Florida governor Jeb Bush, who was perceived as soft on immigration by the Republican right, was one of several candidates Trump defeated in the party’s 2016 presidential primary contest.

Mike Pence, Trump’s vice-president, rejected the verdict of Coulter and other ultra-hardliners.

“It’s not amnesty,” he told Fox News on Sunday. “There’s no permanent status here at all, which is what amnesty contemplates.”

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Trump has threatened to unilaterally declare an emergency to secure funding for the wall if the Democratic-controlled House refuses to approve his plans. Legal analysts have warned that such a drastic move would likely be halted by the courts.

Senator Mark Warner of Virginia, a Democrat, said on Sunday that the chaotic shutdown would only be repeated on future policy disputes if his party caved to Trump’s demands on wall funding before a reopening of the government.

“If the president can arbitrarily shut down the government now, he will do it time and again,” Warner told NBC’s Meet The Press. He and other Democrats have proposed passing separate legislation to ensure the out-of-pocket government workers are paid.

Several opinion polls have indicated that a majority of Americans oppose Trump’s plan to build a border wall and that more people blame Trump for the ongoing shutdown than blame Democrats in Congress.

An average of polls compiled by RealClearPolitics currently states that 55.3% of Americans disapprove of Trump’s performance, the highest figure since last March. A little over 41% of people approve.